An extensive re-evaluation of evidence and analyses of the Randomised Badger Culling Trial II: In neighbouring areas

In the second investigation in a pair of analyses which re-evaluates the Randomised Badger Culling Trial (RBCT), we estimate the effects of proactive badger culling on the incidence of tuberculosis (TB) in cattle populations in unculled neighbouring areas. Throughout peer-reviewed analyses of the RBCT, proactive culling was estimated to have detrimental effects on the incidence of herd breakdowns (i.e. TB incidents) in neighbouring areas. Using previously published, publicly available data, we appraise a variety of frequentist and Bayesian models as we estimate the effects of proactive culling on confirmed herd breakdowns in unculled neighbouring areas. For the during trial period from the initial culls until 4 September 2005, we estimate consistently high probabilities that proactive culling had adverse effects on confirmed herd breakdowns in unculled neighbouring areas, thus supporting the theory of heightened risk of TB for the neighbouring cattle populations. Negligible culling effects are estimated in the post-trial period across the statistical approaches and imply unsustained long-term effects for unculled neighbouring areas. Therefore, when considered alongside estimated beneficial effects within proactive culling areas, these conflicting adverse side effects render proactive culling complex, and thus, decision making regarding potential culling strategies should include (i) ecological, geographical and scientific considerations and (ii) cost–benefit analyses.


Introduction 1.Bovine tuberculosis in Great Britain
Bovine tuberculosis (TB) is an infectious zoonotic disease caused by Mycobacterium bovis.European badgers (Meles meles) have been identified as wildlife hosts for M. bovis.Due to the ecological, public health and economical consequences of bovine TB, badger culling strategies were implemented in attempts to curb transmission of M. bovis and thus reduce the incidence of TB in cattle across Great Britain.
In England, the Randomised Badger Culling Trial (RBCT) was a large-scale randomised field trial conducted in 30 trial areas (of similar size; approximately 100 km 2 ) with high incidence of bovine TB.The RBCT was undertaken to examine the effectiveness of badger culling as a strategy for controlling incidence of TB in cattle in Great Britain.The trial was designed and overseen by a group of independent scientists, collectively known as the independent science group (ISG) on cattle TB [1].The study design (including the ethical, ecological and statistical considerations) and associated culling activities of the RBCT were previously described in our first investigation [2].
The importance and policy relevance of an extensive assessment (of the effects of culling) within culled and unculled neighbouring areas are exemplified by the ongoing debates surrounding recent trends in bovine TB in Great Britain and the effectiveness of badger culling, as discussed in our first investigation [2].

Analyses of neighbouring areas of the RBCT
The statistical analyses of the RBCT were pre-defined (before the incidence data were collected) and also independently audited by a statistical auditor [1,3,4].The response variables for the analyses were the observed (i) total number of herd breakdowns (inclusive of confirmed and unconfirmed) and (ii) number of confirmed herd breakdowns in a trial area.Confirmed herd breakdowns were herd incidents which involved evidence of TB exposure in at least one cattle herd member, and either lesion characteristic of TB was subsequently identified at post-mortem or the M. bovis organism was cultured.Otherwise, breakdowns were classified as unconfirmed.
An interim analysis of the RBCT was presented in a peer-reviewed analysis in 2003 [5].The finding of increased TB incidence in cattle in areas subject to reactive culling led to the suspension of reactive culling.Subsequent ecological studies investigated the estimated increased TB risk induced by reactive culling and observed differences to badger ecology and behaviour as well as M. bovis epidemiology in badgers.In particular, increased ranges and increased mixing, and hence greater M. bovis transmission and prevalence, within badger populations, were estimated as well as increases to the number of cattle herds potentially in contact with each infected badger [6][7][8][9].
The peer-reviewed analyses of the RBCT consistently considered the effects of proactive culling on the incidence of herd breakdowns in the areas neighbouring proactive culling areas (alongside the within-culling areas).Consistent with the inferred effects of reactive culling, a 29% increased risk of TB, relative to herds in lands neighbouring unculled survey-only areas, was observed in cattle herds residing in neighbouring areas of up to 2 km outside the proactive culling areas [10].Note that we analyse the effects of culling within proactive culling areas separately [2].
The ISG final report in 2007, which appraised 55.8 triplet years of the RBCT data, estimated that the beneficial effects of reduced TB incidence within proactively culled areas were offset by increased incidence in cattle in the unculled neighbouring areas [11].Thus, due to the potential for contrasting effects of badger culling in the distinct areas, associated analyses explored the net effects of successive annual culls [12].The estimated adverse effects in neighbouring areas tended to decline upon successive culls of the RBCT, and the estimated net effect was beneficial after the fourth and later annual culls.The reduced adverse effects over time in neighbouring areas were hypothesized to be a consequence of the changes to the badger population induced by repeated systematic culling.In particular, the culling was inferred to likely expand badgers' daily ranges and increase their dispersal (the permanent movement from one location to another).Nevertheless, the short-distance nature of dispersal meant that the repeated culling was thought to be likely responsible for a depleted population of dispersers and achievement of a quasi-stable spatial organization [12].
Following the completion of the RBCT, post-trial analyses (from 1 year after the last cull) assessed whether the latter-stage results of reductions in elevated risk in neighbouring areas were sustained.The detrimental during-trial effects of greater TB incidence in neighbouring areas were estimated to be no longer present in an analysis of the post-trial period [13], and any beneficial post-trial effects disappeared after 18 months of post-culling [14].A complementary analysis estimated that there were no sustained post-trial effects of proactive culling on incidence in the neighbouring areas up to March 2013 [15].
Finally, the findings and concerns from a recent preprint analysis by Torgerson et al. [16] were primarily related to the effects of badger culling within proactive culling areas and were individually addressed by Mills et al. [2].With respect to the effects of culling in unculled neighbouring areas, the authors of [16] reported that, within a restricted period until 4 September 2005, an analysis displayed a lack of sufficient evidence for side effects of culling in the selected studied period and that the Poisson generalized linear model (GLM) approach taken by the ISG was inferior to alternative model formulations.

Objectives of the current analysis
The primary objective of the current analysis is to comprehensively re-assess the available evidence from the RBCT regarding the effects of proactive culling on herd breakdowns in unculled neighbouring areas.Our in-depth re-evaluation directly addresses the cited lack of evidence (proposed in the separate preprint manuscript by Torgerson et al. [16]) for the side effects in cattle populations in neighbouring areas.
We analyse the herd breakdowns data from three study periods (which were discussed in previous analyses of RBCT data): from the initial cull until 4 September 2005, from the first follow-up cull until 4 September 2005 and the post-RBCT trial period (from 1 year after the last cull until March 2013).Separately, our complementary analysis examines the effects of proactive culling within proactive culling areas across the same three study periods [2].
Our extension to a wide array of statistical techniques and study periods means that we extend beyond previous approaches taken by the ISG, by other subsequent analyses [17,18] and by the recent, separate preprint manuscript [16], and we subject each approach to the same rigorous model checking and model comparison.In doing so, we can make robust conclusions regarding the overall effects of proactive badger culling for neighbouring areas which are strongly informed by consistent scientific evidence, irrespective of which of the appropriate statistical approaches are taken.

Datasets and statistical considerations
The source of data for the current analysis is the RBCT, and our studied during-trial periods cover (i) from the initial proactive cull until 4 September 2005 and (ii) from the follow-up proactive cull until 4 September 2005; 4 September 2005 represents the last during-trial date for which trial data were available for the analysis by Donnelly et al. [10], while the recent preprint manuscript also used data from the initial cull until 4 September 2005 [16].The during-trial data amount to totals of (i) 46.6 triplet years since the initial cull and (ii) 34.1 triplet years since the first follow-up cull.The post-trial period consists of 66.6 triplet years, from 1 year after the last cull until 28 March 2013.Our primary variable of interest is the number of confirmed herd breakdowns (or equivalently, the incidence) during the respective studied period in areas neighbouring proactive culling areas and in areas neighbouring survey-only areas (up to 2 km outside the culling areas).
For comprehensiveness, we also analyse incidence data from the post-trial period from 1 year after the last proactive cull until 28 March 2013, using data downloaded in July 2013.Confirmed herd breakdowns are recorded in 6-month time intervals.
In general, due to the statistical properties of our response (herd breakdowns represent count data), we note the potential for the phenomenon of overdispersion.The principles behind the manifestation of overdispersion, alongside various possible statistical approaches used to address the issue, are described at length in the accompanying analysis for within-culling areas [2].

Statistical methodology 2.2.1. Statistical approach of the independent science group
The trial design of the RBCT, the statistical analyses of the independent science group (ISG) and the policies regarding the release of data were pre-defined and independently reviewed by a statistical auditor [1,3].Regular, independent reviews by the statistical auditor continued from 2000 [4] until the ISG final report in 2007 [11,19].
The ISG-led, independently audited statistical analyses for the RBCT data involved the usage of a log-linear Poisson regression which modelled the observed herd breakdowns [1,3,5,10,12,20].The Poisson regression model sought to quantify treatment/culling effects and made adjustments for factors such as individual triplet effects, the log of the number of baseline herds at risk and the log of the historic 3-year incidence of herd breakdowns.Alternative model formulations and sensitivity of inferences were appraised.Other formulations included covariate interactions with treatment effects, different time periods and adjustments for log of the baseline total cattle numbers and log of the total number of tests conducted.Results obtained were similar across each of the different modelling environments.
Our models estimate the effects of proactive culling by drawing comparison between the confirmed herd breakdowns in areas neighbouring proactive culling areas and in areas neighbouring survey-only areas.As previously described in [2], the models considered here enable a comprehensive assessment of the effects of proactive culling on confirmed herd breakdowns in neighbouring areas as the models again adopt differing parametric families (e.g.Poisson versus quasi-Poisson), specifications (e.g.covariate versus offset forms) and approaches to statistical inference (both frequentism and Bayesianism) across three studied periods (two during-trial periods and one post-trial period).The extension to a Bayesian perspective to modelling in each studied period (to make probabilistic statements about the effects of proactive culling) differs to the solely frequentist approaches taken by the ISG.Our separate, complementary analysis explores the number of confirmed herd breakdowns within proactive culling areas [2].

Model appraisal and diagnostics
Across the frequentist and Bayesian settings, our model appraisal generally involves an in-depth assessment of the validity of underlying assumptions used to enable statistical inference.The model diagnostics are described elsewhere in [2], and we include a summary of key diagnostics employed for frequentist and Bayesian models in electronic supplementary material, tables S3 and S4, respectively.

Results
We present results here across frequentist and Bayesian statistical approaches, for data from the initial cull to 4 September 2005 and from the post-trial period (1 year after the last cull until March 2013).Results for incidence data from the first follow-up cull until 4 September 2005 are included in electronic supplementary materials, 4.2 and 5.2.

From the initial cull until 4 September 2005
The following analysis contains results from frequentist models fitted to confirmed herd breakdowns in areas neighbouring proactive culling areas and in areas neighbouring survey-only areas in the period from the initial culls of the RBCT until 4 September 2005.The incidence data analysed here are the same previously published, publicly available data from [10] and were also used in the recent, separate preprint manuscript by Torgerson et al. [16].Table 1a,b contains a summary of our key results, and we subsequently provide an overview of key observations.Sample model diagnostics and checks are displayed in electronic supplementary material, 4.1.

Table 1a.
For confirmed herd breakdowns from the initial cull until 4 September 2005 in areas neighbouring proactive culling areas and in areas neighbouring survey-only areas of the RBCT, we present a range of frequentist models.Various Poisson GLMs were examined across analyses of the RBCT led by the ISG, and a final Poisson GLM was fitted to the incidence data for the period from the initial cull until 4 September 2005 in [10].The model was independently audited by a statistical auditor who certified the accuracy of the findings and the associated interpretations [19,21].In a separate preprint manuscript by Torgerson et al. [16], other variants of the model were proposed, each of which are denoted by an asterisk (*) below.BIC and AICc denote the Bayesian information criterion and small-sample corrected Akaike information criterion, respectively.Lower values of each information criterion are better but note that it is not appropriate to compare the information criteria of the normal linear model (which assumes a continuous response and assumes normality of errors in a different model fitting method) with the Poisson-based models.From table 1a,b, we deduce that across a range of model fits, there is consistently an estimated adverse effect of proactive culling on the incidence of confirmed herd breakdowns in neighbouring areas.In three of the five GLMs which consider the side effects of proactive culling, the estimated effect is statistically significant.The two model exceptions are difficult to validate as being appropriate due to assumptions such as a highly flexible quasi-Poisson model structure or a possibly constraining assumption of an offset specification.Indeed, such model forms may induce potentially inappropriate model-based uncertainty or problematic diagnostic flaws.
No single model fit outperforms across all information criteria and predictive accuracy metrics.Nevertheless, the best-fitting GLM (model 8) in terms of several information criteria (such as corrected Akaike information criterion (AICc) which measures predictive capabilities) and several leave-one-out predictive accuracy metrics (which approximate model generalizability) is a model which does not account for any effect of culling on confirmed herd breakdowns in neighbouring areas.However, despite the representative fit of the model without any modelled culling effect, other model fits (such as the original Poisson GLM of [10] in generalized Poisson form) attain superior values for other information criteria such as Bayesian information criterion (BIC) (which is a measure of goodness of fit unlike Akaike information criterion and AICc which measures predictive accuracy).The results of the frequentist analysis for the post-trial period differ from the during-trial results as consistently, the estimated effects of culling in unculled neighbouring areas are negligible, with 95% CIs for the estimated effects concentrated symmetrically around small values near zero.
For instance, among the models which estimate the effect of proactive culling, under the best-fitting model (in terms of leave-one-out predictive accuracy and information criteria such as AICc and BIC), the estimated culling effect is marginally beneficial and not statistically significant (estimate: −3.0%, 95% CI: −16.7%, 12.9%).
Indeed, no model fit for the post-trial period yields a significant (either beneficial or detrimental) estimated effect of proactive culling.Hence, we observe superior model fits for model structures which

Bayesian approaches to modelling
To enhance the robustness of model-based inferences and reduce sensitivity to the modelling approach used, we now present results from Bayesian models fitted to confirmed herd breakdowns in areas neighbouring proactive culling areas and in areas neighbouring survey-only areas.

From the initial cull until 4 September 2005
Table 2a,b captures the key results from Bayesian models for the period from the initial cull until 4 September 2005, and we subsequently summarize observations from across the models.In figure 1, we visualize the implied effect of culling for neighbouring areas under the best-fitting Bayesian negative binomial and Poisson GLMs.Noteworthy sample model diagnostics and checks are displayed in electronic supplementary material, 5.1.Posterior predictive-based diagnostics enable identification of systematic model misspecification, comparison of model fits and, hence, conclusions regarding model appropriateness using varying forms of likelihood and varying prior distributions.Under the conditions of various Bayesian GLMs with weakly informative, regularizing prior distributions, there are high probabilities of between 93.2% and 98.3% that proactive culling has an adverse effect on the incidence of confirmed herd breakdowns in areas neighbouring proactive culling areas and in areas neighbouring survey-only areas.Potential model misspecification is implied by the fitted negative binomial models from the separate preprint manuscript by Torgerson et al. [16].Hence, focusing on our proposed improved models which yield greater estimated generalizability, under the best-fitting negative binomial GLM (Model c.2), there is a 94.0% probability that proactive culling had an adverse effect on confirmed herd breakdowns in neighbouring areas.
Interestingly, a model (Model d.2) without any explicit consideration for an effect of culling does not yield any discernible model diagnostic flaws, which is perhaps reflective of uncertainty regarding the extent of the effect of proactive culling on herd breakdowns in neighbouring areas.Nevertheless, the findings are consistent with those of [10] as the range of best-fitting models implies that it is highly probable that proactive culling resulted in increased incidence in adjoining or neighbouring areas in the period from the initial proactive cull until 4 September 2005.royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 240386

Post-trial period
We fitted the Bayesian models to confirm herd breakdowns in the post-trial period, from 1 year after the last proactive cull until 28 March 2013.Individual model appraisals are presented in electronic supplementary material, table S8a-c, where the same suite of model diagnostics are employed, identical to those performed for the two during-trial study periods.Figure 2 illustrates the implied culling effects under the best-fitting Bayesian negative binomial and Poisson GLMs.Similar to our frequentist-based analysis, across a range of Bayesian models for the post-trial period (electronic supplementary material, table S8a-c) [22][23][24], we estimate small and negligible post-trial effects of proactive culling on confirmed incidence in neighbouring areas.In particular, under the conditions of any of the models, the estimated 95% credible intervals (CrIs) for the effects of culling are uncertain and almost symmetric around zero.Among the subset of models which explicitly account for any effects of proactive culling, under the best-fitting model (figure 2), there is a 63.2% probability (Model c.2) that proactive culling had a detrimental effect on confirmed herd breakdowns in neighbouring areas, yet there is a 74.5% probability that the adverse effect was less than a 15% increase in confirmed herd breakdowns.
Indeed, as a consequence of the estimated negligible effects of proactive culling on confirmed herd breakdowns, we again find (consistent with the separate frequentist analysis for the post-trial period) that a model without any consideration for the effects of proactive culling (Model d.2) achieves the best (estimated) out-of-sample predictive accuracy.The superiority of a model structure without a model proactive culling effect is apparently due to the unsustained nature of any significant culling effects on herd breakdowns in neighbouring areas.

Discussion
We have comprehensively re-assessed the available evidence regarding the effects of proactive culling on the incidence of herd breakdowns for cattle residing in unculled neighbouring areas.Similar to our accompanying analysis [2], our modelling re-evaluation covers three study periods (two duringtrial and one post-trial), and across two distinct approaches to statistical inference (frequentism and Bayesianism), a wide range of model frameworks (identical to [2] to ensure consistency) are analysed.
In the period from the initial cull until 4 September 2005 (the period previously analysed by [10] and discussed in the recent separate preprint manuscript by [16]), across various appropriate frequentist models, we estimated adverse effects of proactive culling on the incidence of confirmed herd breakdowns in areas neighbouring culled areas.The implied adverse effect was consistently significant across several well-fitting models, and the finding aligns with the conclusions of various peer-reviewed analyses.The estimated adverse effect was subject to a degree of modelling uncertainty From the posterior distributions, we deduce that under the negative binomial and Poisson models' conditions, there are 63.2% and 45.3% probabilities, respectively, that the proactive culling effect on confirmed incidence was adverse, yet there are high probabilities of 74.5% and 98.2%, respectively, that the effect was less than a 15% increase on confirmed incidence.
Table 2a.For confirmed herd breakdowns from the initial cull until 4 September 2005 in areas neighbouring proactive culling areas and in areas neighbouring survey-only areas of the RBCT, we present a range of Bayesian models.The original, frequentist Poisson GLM used in [10] was re-specified in the Bayesian paradigm in the separate preprint manuscript by Torgerson et al. [16], with an alternative negative binomial likelihood (as opposed to Poisson likelihood), and is labelled as Models a.1 and a.2 below.Models discussed in the separate preprint manuscript by Torgerson et al. [16] are denoted by an asterisk (*), and our improved/alternative model versions are denoted by two asterisks (**).With respect to the preprint manuscript by Torgerson et al. [16], we employ a different labelling system for models, and the corresponding label from the preprint manuscript can be found in electronic supplementary material, table S2.Estimated culling effects and associated uncertainty are reported using the posterior median and 95% credible intervals (CrIs) of exponentiated posterior samples of the models' culling effect parameter.as a model structure without explicit consideration of the effects of proactive culling achieved a representative fit (in terms of some out-of-sample predictive accuracy metrics and some information criteria).Therefore, our extension to a separate Bayesian analysis allowed us to directly address any uncertainty regarding the effects of proactive culling for unculled neighbouring areas.The Bayesian models consistently assigned high probabilities to adverse effects of culling on herd breakdowns in areas neighbouring proactively culled areas, relative to areas which neighbour survey-only areas.For instance, under the conditions of the best-fitting, most generalizable Bayesian model (Model c.2), there is a 93.2% probability that proactive culling had an adverse effect on confirmed herd breakdowns in neighbouring areas.
Expanding our focus to the period from the first follow-up proactive cull until 4 September 2005, irrespective of the distinct approaches to statistical inference (either frequentist or Bayesian models), we again estimated with high probabilities that there were detrimental side effects of proactive culling for cattle in neighbouring areas.Nevertheless, the best-fitting frequentist model fit (for the period from the first follow-up cull until 4 September 2005) was attained for a structure which did not  [10] was re-specified in the Bayesian paradigm in the separate preprint manuscript by [16], with an alternative negative binomial likelihood (as opposed to Poisson likelihood), and is labelled as Models a.1 and a.2.Models discussed in the separate preprint manuscript by Torgerson et al. [16] are denoted by an asterisk (*), and our improved/alternative model versions are denoted by two asterisks (**).With respect to the preprint manuscript by Torgerson et al. [16], we employ a different labelling system for models, and the corresponding label from the preprint manuscript can be found in electronic supplementary material, table S2.Estimated culling effects and associated uncertainty are reported using the posterior median and 95% credible intervals (CrIs) of exponentiated posterior samples of the models' culling effect parameter.account for the effect of proactive culling.The model fit was superior across a range of out-of-sample predictive metrics and information criteria, which is perhaps a consequence of decreasingly substantial side effects of proactive culling in a restricted study period that focuses more heavily on the latter stages of the during-trial period.Our Bayesian analysis still assigned high plausibility to the event that proactive culling had adverse effects for cattle populations in neighbouring areas, albeit estimating weaker overall effects.The weaker detrimental effects of proactive culling (for cattle in neighbouring areas) from the first follow-up cull until 4 September 2005 (relative to the period from the initial cull until 4 September 2005) had been observed by separate, peer-reviewed analyses of the RBCT which employed different statistical methods [10,12].The phenomenon is apparently a consequence of the net beneficial effects of individual, latter-stage proactive culls, in contrast to the marked net detrimental effect between the initial and first follow-up cull [12].
In the post-trial period, our range of frequentist models consistently estimated negligible (nonsignificant) and, thus, non-existent effects of proactive culling on herd breakdowns for unculled neighbouring areas.Indeed, the best-fitting, most generalizable models were consistently those that did not explicitly consider any post-trial effect of proactive culling for the neighbouring areas.The findings of non-existent and, thus, unsustained effects of proactive culling in neighbouring areas were substantiated and quantified more precisely by our separate Bayesian model-based estimates of high probabilities for limited post-trial effects of culling.
Therefore, based on our comprehensive analyses, we deduce that during the RBCT, irrespective of statistical methodology, proactive culling was associated with a heightened risk (albeit declining with time) of herd breakdowns for cattle residing in areas neighbouring culled areas, compared to cattle populations in areas neighbouring unculled survey-only areas.The adverse side effects of proactive culling, particularly in the immediate aftermath of the first proactive cull, represent important considerations for scientists and policy-makers during the design and implementation of any potential  1) badger culling strategy.Consequently, it may be required to perform comprehensive assessments of the geographies of any planned culling areas and their environs.Furthermore, our analysis reveals that the side effects of proactive culling for neighbouring areas declined over time and were unsustained following the completion of annual culls, thus apparently removing concerns regarding long-term effects in the absence of concurrent proactive culls.Indeed, the estimated during-trial adverse effects for neighbouring areas were substantiated by the absence of adverse effects in neighbouring areas when proactive culling was removed.Therefore, we deduce that any proactive culling strategies require careful consideration for neighbouring areas during culling due to adverse effects between annual culls, yet in contrast to the post-trial findings for within-culling areas, post-trial considerations are apparently not as important for neighbouring areas as there are no long-term effects beyond the final annual cull.
Combining the analysis with the beneficial during-trial and post-trial effects for cattle populations within proactive culling areas [2], we conclude that development of any proactive culling strategy is likely to remain controversial and require in-depth ecological and scientific considerations.It would be instructive to carefully consider the estimated benefits to cattle residing within proactive culling areas alongside the consideration of risk mitigation strategies for neighbouring cattle populations.A detailed cost-benefit analysis would be a suitable avenue to scrutinize such contrasting estimated benefits and risks to proactively culled and unculled neighbouring areas, respectively.

Figure 1 .
Figure 1.The posterior distribution, alongside 95% CrI, of the treatment effect from the best-fitting Bayesian negative binomial (Model c.2; green) and Poisson (Model e; yellow) GLMs fitted to confirmed herd breakdowns for the period from the initial cull until 4 September 2005.The vertical dashed line located at zero is reflective of hypothetical absence of any culling effect.From the posterior distributions, we deduce that under the models' conditions, there are probabilities of 93.2% and 98.3%, respectively, that the proactive culling effect on confirmed incidence was detrimental to unculled neighbouring areas in the period from the initial cull until 4 September 2005.

Figure 2 .
Figure2.Among the models which explicitly consider culling effects, the posterior distribution, alongside 95% CrI, of the treatment effect parameter from the best-fitting Bayesian negative binomial (Model c.2; green) and Poisson (Model e; yellow) GLMs fitted to confirmed herd breakdowns for the post-trial period (from 1 year after the last proactive cull until 28 March 2013).The vertical dashed line located at zero is reflective of hypothetical absence of any culling effect.From the posterior distributions, we deduce that under the negative binomial and Poisson models' conditions, there are 63.2% and 45.3% probabilities, respectively, that the proactive culling effect on confirmed incidence was adverse, yet there are high probabilities of 74.5% and 98.2%, respectively, that the effect was less than a 15% increase on confirmed incidence.

Table 1b .
[16]21]firmed herd breakdowns from the initial cull until 4 September 2005 in areas neighbouring proactive culling areas and in areas neighbouring survey-only areas of the RBCT, we present a range of frequentist models.Various Poisson GLMs were examined across analyses of the RBCT led by the ISG, and a final Poisson GLM was fitted to the incidence data for the period from the initial cull until 4 September 2005 in[10].The model was independently audited by a statistical auditor who certified the accuracy of the findings and the associated interpretations[19,21].In a separate preprint manuscript by Torgerson et al.[16], other variants of the model were proposed, each of which are denoted by an asterisk (*) below.BIC and AICc denote the Bayesian information criterion and small-sample corrected Akaike information criterion, respectively.Lower values of each information criterion are better but note that it is not appropriate to compare the information criteria of the normal linear model (which assumes a continuous response and assumes normality of errors in a different model fitting method) with the Poisson-based models.We fitted the frequentist models to the confirmed herd breakdowns from the post-trial period, from 1 year after the last cull until 28 March 2013.Individual model appraisals are presented in electronic supplementary material, table S6a,b.

Table 2b .
For confirmed herd breakdowns from the initial cull until 4 September 2005 in areas neighbouring proactive culling areas and in areas neighbouring survey-only areas of the RBCT, we present a range of Bayesian models.The original, frequentist Poisson GLM used in